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Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equations
Vikas Dwivedi, Balaji Srinivasan
TL;DR
The paper addresses limitations of mesh-based PDE solvers and deep PINNs for solving linear PDEs, especially in complex domains and large-domain settings. It develops PIELM, a rapid physics-informed extreme learning method, and DPIELM, its distributed extension. PIELM and DPIELM show strong performance across stationary and time-dependent problems, while DPIELM produces results comparable to conventional numerical techniques.
Problem
Complex-domain PDEs challenge mesh-based methods, while PINNs lack clear architecture and data-size guidance, can encounter local minima, and learn slowly.
Method
PIELM combines ELM with PINN physics constraints for stationary and time-dependent linear PDEs, and DPIELM extends representation without extra hidden layers.
Results
PIELM and DPIELM exhibit superior performance across stationary and time-dependent problems compared with existing methods.
Takeaways & Limitations
DPIELM produces results comparable to conventional numerical techniques for time-dependent problems, supporting neural-network PDE solution in complex domains.
Takeaways & Limitations
PIELM has limited representation capacity for complex functions, including profiles with sharp gradients and corners.
Abstract
from arXiv · showhide
There has been rapid progress recently on the application of deep networks to the solution of partial differential equations, collectively labelled as Physics Informed Neural Networks (PINNs). In this paper, we develop Physics Informed Extreme Learning Machine (PIELM), a rapid version of PINNs which can be applied to stationary and time dependent linear partial differential equations. We demonstrate that PIELM matches or exceeds the accuracy of PINNs on a range of problems. We also discuss the limitations of neural network based approaches, including our PIELM, in the solution of PDEs on large domains and suggest an extension, a distributed version of our algorithm -{}- DPIELM. We show that DPIELM produces excellent results comparable to conventional numerical techniques in the solution of time-dependent problems. Collectively, this work contributes towards making the use of neural networks in the solution of partial differential equations in complex domains as a competitive alternative to conventional discretization techniques.
1 Introduction
Mesh-based PDE solvers face difficulties with complex geometries, discretization discrepancies, and the computational cost of deep PINNs. The paper introduces PIELM, combining physics-informed learning with the fast ELM framework, and proposes DPIELM to improve representation without extra hidden layers.
- Mesh-based methods can become infeasible on complex computational domains because grid generation is difficult.
- Discretization creates a discrepancy between the mathematical PDE and its approximate difference equation, sometimes causing serious problems.
- Neural-network PDE solvers are meshfree and can incorporate governing equations and boundary conditions into training.
- PINNs lack theoretical guidance for architecture and data size, may encounter local minima, and learn more slowly than traditional numerical methods.
- PIELM combines ELM and PINN by using fixed hidden-layer weights and incorporating PDE physics into the cost function.
- PIELM and DPIELM are evaluated on stationary and time-dependent PDEs, with DPIELM extending representation without adding hidden layers.
3 Proposed PIELM
PIELM solves linear stationary and time-dependent PDEs by enforcing PDE, boundary, and initial-condition constraints through a shallow network and linear algebra. Its direct formulation determines output weights with a pseudoinverse rather than iterative optimization.
- PIELM approximates a linear unsteady PDE solution while enforcing boundary and initial conditions over the computational domain.
- The formulation uses the Moore–Penrose generalized inverse, which works for singular and nonsquare H matrices.
- The method uses a shallow network with randomly assigned input-layer weights and forms equations for PDE, boundary-condition, and initial-condition errors.
- The three equation sets are assembled as Hc = K, and the output-layer weights are computed as pinv(H)K.
- Unlike conventional ELM minimization, PIELM uses a direct approach that imposes the physics in an exact sense.
4 Performance evaluation of PIELM
PIELM is evaluated on stationary and time-dependent linear PDEs, including complex geometries and advection-diffusion cases. It often achieves high accuracy with fewer points and rapid learning, but struggles with sharp gradients and some unsteady problems.
- PIELM is tested on stationary and time-dependent linear and quasi-linear PDEs, with errors measured against exact solutions.Experiments were conducted in Matlab 2017b on an Intel Core i5 laptop with 8GB RAM.
- 1D steady cases: 100 points yielded 10^-5 and 10^-3 accuracy for the unified deep ANN in TC-1 and TC-2, whereas PIELM used fewer than half the points and reached 10^-4 in both cases.
- 2D steady cases: On complex 2D geometries, PIELM solved advection and diffusion cases on Ω1 and a detailed Illinois-shaped polygon on Ω2.The Illinois-shaped case used 2370 points and achieved 10^-7 accuracy.
- 2D steady cases: 5500 points produced 10^-3 accuracy for the unified deep ANN in TC-4 and TC-5, while PIELM used 1161 points and achieved 10^-6 and 10^-4, respectively.
- Unsteady advection: For unsteady 1D advection, PIELM correctly predicts constant- and variable-coefficient solutions using 420 collocation, 21 boundary, and 20 initial-condition points within 2–3 seconds.The method also permits larger time steps than traditional upwinding schemes, which are restricted by stability conditions.
- Unsteady advection-diffusion: For unsteady advection-diffusion, PIELM captures advection speed and bounded oscillations, but can violate initial and boundary conditions, produce oscillations, and model Gaussian decay incorrectly.Increasing the hidden layer to 7780 neurons did not improve the poor prediction; the 2D case used 125000 data points and 1000 neurons, with improvement requiring still more points.
4. It is meshfree method and can handle the complex geometries.(TC-6)
PIELM is evaluated on complex and sharply varying PDE solutions, where its representation limitations become apparent. A deep PINN also fails to capture a high-frequency wavelet.
- PIELM reduces arbitrariness in selecting the hidden-layer neuron count.
- PIELM’s representation of discontinuities and sharp gradients is tested using a composite 1D function and a sharply peaked 2D Gaussian.
- The tests expose PIELM’s limitation in representing profiles with sharp gradients and corners.
- PIELM fails on PDEs whose exact solutions contain sharp gradients, including sharp-peaked advection and low-diffusion convection–diffusion cases.
- A deep PINN with 9 hidden layers and 20 neurons per layer also cannot capture the sharp gradients of a high-frequency wavelet.
- The paper summarizes PIELM’s main limitation as inability to represent complex functions, restricting it on PDEs with sharp exact solutions.
6 Distributed PIELM
DPIELM distributes PIELM across non-overlapping cells, using local representations and interface constraints to handle complicated functions more effectively.
- DPIELM partitions the computational domain into multiple cells and installs a separate PIELM in each cell.
- The method is motivated by finite volume methods, stitching cell solutions through additional convective and diffusive flux conditions at interfaces.
- Each local PIELM uses a different representation in its domain portion while satisfying continuity and differentiability constraints across cell interfaces.
- For advection, continuity of the solution is sufficient; diffusion problems require the solution to be continuously differentiable.
- Implementation begins by uniformly dividing the domain into non-overlapping cells and installing one PIELM in each cell.
- DPIELM assembles cellwise PDE, boundary, initial, and interface equations into a linear system Hc = K and obtains c using a pseudoinverse.
7 Performance evaluation of DPIELM
DPIELM is evaluated on cases where regular PIELM and PINN failed, including nonsmooth, sharply peaked, and unsteady advection–diffusion problems. The reported figures show exact and DPIELM solutions across these tests.
- DPIELM is tested on all cases in which regular PIELM and PINN failed, using architectures specified in Table 5.
- DPIELM results for nonsmooth 1D functions and sharply peaked 2D Gaussians are compared with exact solutions in Figures 22 and 23.
- DPIELM solutions for sharp-peaked Gaussian advection and high-frequency wave-packet advection are reported in Figures 25 and 26.
- The 1D and 2D unsteady convection–diffusion results are shown in Figures 27–30 at multiple time points.
- For unsteady 1D and 2D convection–diffusion, DPIELM equations include time, advection, and diffusion derivative terms.
- Partitioning simplifies representation in individual cells, enabling local PIELMs to capture sharp gradients and discontinuities in the reported tests.
- The paper reports a first ELM-based demonstration for 2D unsteady PDEs and states that DPIELM outperforms a deep PINN in representation ability.
8 Conclusion and future work
PIELM is presented as an efficient method for linear PDEs, while DPIELM addresses whole-domain representation limits by distributing representations across domain portions. The authors identify large-domain and nonlinear problems, and the need for more rigorous comparisons, as future work.
- PIELM solves stationary and time-dependent linear PDEs while respecting PDE physics and supporting complex geometries through a meshfree approach.
- PIELM reduces numerical artefacts such as false diffusion and, for fixed problems, is reported as more accurate and faster than prior deep neural-network results.
- DPIELM uses different representations across domain portions while imposing continuity and differentiability constraints to address single-network limitations on complex functions.
- The authors state that a firm conclusion about competitiveness with conventional techniques requires a more thorough and fair study.
- The present study is limited to linear problems because PIELM’s efficacy depends on the linear nature of the final problem.